1. This question is about the terminology that we use in econometrics.  Suppose the relationship between y and x is given by:

y = β₀ + β₁x + u.

Using a data sample, you calculate the OLS estimates β^{^}₀ and β^{^}₁, resulting in the following  OLS regression line (sample regression function):

y^{^} = β^{^}₀ + β^{^}₁x.

The difference between y and y^{^} is defined as:

y^{^} -  Y = u^{^}

What do we call the following variables? (each can be answered in 1-3 words):

y = β₀ + β₁ + u

a. y               e. u^{^}

b. x               f. β₀

c. u               g. β₁

d. y^{^}

2. You want to see how differences in fuel efficiency affect the price of a car.

a. Calculate the OLS estimates β^{^}₀ and β^{^}₁ for the following model using the sample of 11 cars (you must show some of your work):

price = β₀ + β₁mpg + u

i   price   mpg

1   3,799    22

2   3,299    29

3   3,667    24

4   3,955    19

5   3,984    30

6   3,829    22

7   3,291    20

8   3,995    30

9   3,895    26

10  3,798    35

11  3,748    31

b. These data are a subset of a dataset that comes preloaded with every copy of Stata.  Type the following commands in this order to load the data into Stata:

clear all

sysuse auto

keep if price<4000

keep price mpg

Once the data is loaded, run a regression of price on mpg using the following command:

regress price mpg

Do the Stata estimates agree with your calculation?  Copy and paste the

results from just this last command into your assignment.

3. You are studying the relationship between GNP and life expectancy.  Using cross- sectional data set containing the GNP per capita (gnppc) in dollars and the average life expectancy (lexp) in years, you run OLS on the following model:

lexp  = β₀ + β₁gnppc + u

To do so you run the following commands in Stata:

clear all

sysuse lifeexp

regress lexp gnppc

a. Interpret the OLS estimate of the slope parameter β^{^}₁.

b. Suppose you have another variable that measures per capita GNP in thousands of dollars, gnppc1000.  What would be the OLS estimate of the slope parameter from the same model but using this new measure of GNP?

lexp = β₀ + β₁gnppc1000

[Note: 1000 =gnppc/1000]

4. Let kids denote the number of children ever born to a woman, and let educ denote years of education for the woman. A simple model relating fertility to years of education is:

kids = β₀ + β₁educ + u

where u is the unobserved error.

a. What kinds of factors are contained in u? Are these likely to be correlated with education?

b. Will a simple regression analysis uncover the ceteris paribus effect of education on fertility? Explain.

5. Suppose that you are asked to conduct a study to determine whether smaller class sizes lead to improved student performance of fourth graders.

a. If you could conduct any experiment you want, what would you do? Be specific.

b. More realistically, suppose you can collect observational data on several thousand fourth graders in a given state. You can obtain the size of their fourth grade class and a standardized test score taken at the end of fourth grade. Why might you expect a negative correlation between class size and test score?

c. Would a negative correlation necessarily show that a smaller class size causes better performance? Explain.

6. You are studying the relationship between job experience and wages for the population of women able to work in the U.S. Using a cross-sectional data set containing total experience in years (ttl_exp) and hourly wages (wage) for a representative sample of women in the U.S. you estimate the following model using OLS.

wage = β₀ + β₁ttl_exp + u

To do so you run the following commands in Stata in the order listed below:

clear all

sysuse nlsw88.dta

regress wage ttl_exp

a. Copy and Paste your regression results only. Interpret estimated coefficient β^{^}₁

b. How much more per hour is a woman expected to make after she gains 5 more years of experience?

c. You think that the relationship between wages and experience is not linear, but that each year of experience increases hourly wage by a constant percentage. Using the same cross-sectional data you estimate the following model using OLS.

In_wage = β₀ + β₁ttl_exp + u

where ln_wage is just the natural log of wage.

 To do so you run the following commands in Stata in the order listed below:

clear all

sysuse nlsw88.dta

gen ln_wage=log(wage)

regress ln_wage ttl_exp

d. Copy and paste the regression results. What is the interpretation of β^{^}₁ now?